УДК 620.179.16
MAXIMUM KURTOSIS PRINCIPLE FOR THE PARAMETER SELECTION OF GABOR WAVELET AND ITS APPLICATION TO ULTRASONIC SIGNAL PROCESSING
Han Yong, Chen Guo-guang School of Mechatronic Engineering, North University of China, Taiyuan, China
ИСПОЛЬЗОВАНИЕ ПРИНЦИПА МАКСИМАЛЬНОГО ЭКСЦЕССА ДЛЯ ВЫБОРА ПАРАМЕТРА ВЕЙВЛЕТА ГАБОРА И ЕГО ПРИМЕНЕНИЕ ДЛЯ ОБРАБОТКИ УЛЬТРАЗВУКОВЫХ
СИГНАЛОВ
Хай Йонг, Чен Гуо-гуанг Школа мехатроники, Северный китайский университет, Тайюань, Китай
Методы подавления шума с использованием вейвлетного преобразования (ВП) широко применяются в неразрушающем контроле, особенно ультразвуковом. Однако вейв-летные фильтры обладают свойством равенства Q-фактора, так что невозможно одновременно выбрать произвольно центральную частоту и ширину полосы фильтра. В работе развит новый метод использования ВП, позволяющий устранить этот недостаток. Предложен метод идентификации слабых у. з. сигналов с использованием вейвлетного преобразования на основе вейвлета Габора с оптимальным параметром. Выбрана оптимальная центральная частота и ширина полосы вейвлета Габора с использованием алгоритма максимизации эксцесса. Центральная частота и ширина полосы вейвлета Габора хорошо согласуются с соответствующими параметрами у. з. сигнала. Результаты численного моделирования и данные эксперимента представлены для того, чтобы оценить эффективность применения вейвлетного преобразования на основе вейвлета Габора с оптимальным параметром для обнаружения дефекта с использованием ультразвука. Данная методика является простой и эффективной процедурой для обработки сильно зашумлен-ных у. з. сигналов.
Abstract
The noise suppression techniques with wavelet transform (WT) are widely used in nondestructive testing and evaluation (NDT&E), especially in ultrasonics. But the wavelet based filter has the property of equal Q-factor, so, it is impossible to choose the central frequency and the bandwidth arbitrarily at the same time. This paper develops a new technique using WT to eliminate this drawback. In this paper, a weak ultrasonic signals identification method by using the optimal parameter Gabor wavelet transform is proposed. We address the choice of the optimal central frequency and bandwidth of the Gabor wavelet using the kurtosis maximization algorithm. The central frequency and bandwidth of the optimal parameter Gabor wavelet matched that of the ultrasonic signal very well. Numerical and experimental results have been presented to evaluate the effectiveness of the optimal parameter Gabor wavelet transform on ultrasonic flaw detection. This technique is a simpler and effective technique for processing heavy noised ultrasonic signals.
Keywords: NDT&E; Gabor wavelet; wavelet transform; kurtosis.
1. Introduction
The ultrasonic pulse echoes reflected from inhomogeneities or discontinuities in tested materials or specimen contain a large amount of information of the reflectors. These signals are also contaminated by noise originating from both the measurement system and specimen. The noise embedded in useful signals, sometimes even very heavy, places a fundamental limit on the detection of small defects and the accuracy of measurement. The amplitude of the noisy flaw echo may be quite small and so it is difficult to extract the echo with an acceptable signal to noise ratio (SNR). It is essential, therefore, to employ advanced signal enhancement techniques to extract useful diagnostic information from the measured ultrasonic NDE (non destructive evaluation) signals [1,2].
The WT (wavelet transform) is defined in terms of basis functions obtained by compression/dilation and shifting of a mother wavelet [3]. It acts as a mathematical microscope which allows one to zoom in on the fine structure of a signal, or, alternatively, to reveal large scale structures by zooming out. Its property of self-adjusting the analysis windows according to the signal frequency makes it more suitable for the analysis of transient non-stationary ultrasonic signals. In ultrasonic NDT&E, it is not a new concept to use wavelet based signal processor to remove the noise. Various techniques have been utilized for this purpose. In the field of ultrasonic signal de-noising, ultrasonicpulse-echo liked wavelet, a kind of band pass filter to some extent attracts more and more attention.
Abbate et al. [4] first pioneered the technique. In their study, a mother wavelet was selected as an ultrasonic pulse echo, then compresses or expands the scale of the base wavelet to obtain a bank of filters to overlap the ultrasonic frequency band. But the method of selecting the central frequency of both wavelet and actual ultrasonic signal, and the number of band pass filters were not mentioned. Subsequent scholars, like Zhang et al. [5] proposed a method of using the filter with optimal frequency-to-band ratio of wavelet to de-noise the ultrasonic signals. Another scholar, Song et al. [1] designed a wavelet based band-pass filter with its central frequency matched to the actual ultrasonic signals, and proposes the matched rules between them. But, this method is inconvenient for the selection of the appropriate wavelet scale, and must be based on the experience of scholar for wavelet design.
Recently, Liang et al. [6]addressed maximum negentropy to select the optimal scale of wavelet, and the central frequency of the optimal scale wavelet match that of the actual ultrasonic signals well, because they only adjust one parameter (scale of wavelet), their bandwidth did not match well. In another work, Liang and collaborator use maximum nongaussianity principle to design an optimal parameter filter to match the actual ultrasonic signals, in the proposed method, they adjust three parameters to make the optimal parameter filter matching the actual ultrasonic signals completely. In ultrasonic signal processing, the Gabor wavelet which is modulated at a central frequency plays an important role [7, 8], such a wavelet is well correlated with the ultrasonic echoes by adjusting the central frequency and bandwidth, and it has the similar function envelope with the pulse ultrasonic signal, so as to obtaining higher WT coefficients.
The remainder of this paper is organized as follows. In section 2, we address the choice of the optimal central frequency and bandwidth of the Gabor wavelet using the kurtosis maximization algorithm. In section 3 and section 4, results obtained from the analysis of numerical and actual ultrasonic signals are presented. Finally, conclusions are presented in section 5.
2. Theoretical analysis
Ultrasonic pulse detection consists of determining the presence or absence of a pulse and estimating its amplitude and arrival time. In ultrasonic testing, the measured ultrasonic signal f(t) can be expressed as the sum of two components:
ft) = h(t) + n(t). (1)
Where h(t) is the acoustic signal embedded in the noise n(t). The signal f(t) is thus band-limited, corrupted, and distorted by white Gaussian or random noise n(t), n(t) is a broad band signal with its spectra density function covers almost the whole analyzed frequency range. So complete removing of the noise is impossible so far. The best way of noise suppression is maintaining the useful informa-
tion as possible as we can in terms of the testing purpose. The purpose of this work is to obtain a signal fw(t) as close as possible to h(t), thus minimizing the effect of n(t). The ability of adapting the window size in frequency domain makes the WT a natural candidate for the analysis of transient ultrasonic signals with a wide spectral range.
A family of time-frequency atoms 9u,s(t) is obtained by scaling the mother wavelet 9(t) by s, and translating by u:
9u,s(t) = Vj' (2)
s
The wavelets transform of f(t) at time u and scale s is
Wf(u, s) = <f, C> = Jf (t)js 9* [j dt = = J(h (t) + n (t))9* {t—^j dt = Jh (t) 9* ^ dt +
1 in.
+ J n (t) 9* ^j dt = Wh (u, s) + Wn (u, s). (3)
Wh(u, s) is an inner product integral and can be rewritten in the frequency domain by way of Parseval's Identity:
Wh(u, s) = <h, 9u,s> = <H(o>), Vu,s(®)>, (4)
where yu,s(ffl) = -/7y(sffl) e j2nu®, y(ffl) is the Fourier transform of the
wavelet 9(t). The energy of at a particular scale s, and translation u, is given by its squared magnitude [12]
IWh(u, s)l2 = i<H(ffl), yu,s(o>)>I2. (5)
If the central frequency the ultrasonic pulse h(t) is equal to that of the daughter wavelet ysu(t), and the frequency band of the ultrasonic pulse is within but as close as possible to the support of the daughter wavelet, that is to say, majority of the interest frequency components can pass through. Meanwhile the noise beyond this range is well cut down.
The recovered signal fw(n) will be as close as possible to the ultrasonic pulse h(t). It also indicates that one filter is enough to maintain the useful information of the ultrasonic signal. In practice, it is very difficult to select the optimal scale of wavelet, furthermore, though the optimal scale can be selected accurately, unfortunately, it only make the central frequency of the optimal scale wavelet match that of the actual ultrasonic signals well, and their bandwidth do not match well, because the wavelet based filter has the property of equal ^-factor, that is to say when the central frequency of a certain daughter wavelet is determined, its frequency bandwidth is determined too.
This drawback can be eliminated effectively by select two parameters: the central frequency and bandwidth. The Gabor wavelet is a suitable candidate. In ultrasonic signal processing, the Gabor wavelet which is modulated at a central frequency fg plays an important role [7, 8], such a wavelet is well correlated with the ultrasonic echoes, and it has the similar function envelope with the pulse ultra-
sonic signal, so as to obtaining higher WT coefficients. The normalized Gabor wavelet can be expressed as:
g(t) =
1
(a2 n)
1/4
exp
' t2 Л
2a2
cos
(2 ft),
(6)
where a concerns with the bandwidth of the m
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